ABC conjecture: Difference between revisions
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The abc conjecture asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed <math>c^{1-\varepsilon}</math> for any fixed <math>\varepsilon > 0</math> (if a,b,c are smooth). | The '''abc conjecture''' asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed <math>c^{1-\varepsilon}</math> for any fixed <math>\varepsilon > 0</math> (if a,b,c are smooth). | ||
This shows for instance that <math>(1-\varepsilon) \log N / 3</math>-smooth a,b,c of size N which are coprime cannot sum to form a+b=c. This unfortunately seems to be too weak to be of much use for the [[finding primes]] project. | This shows for instance that <math>(1-\varepsilon) \log N / 3</math>-smooth a,b,c of size N which are coprime cannot sum to form a+b=c. This unfortunately seems to be too weak to be of much use for the [[finding primes]] project. | ||
* [[wikipedia:Abc_conjecture|Wikipedia page for the ABC conjecture]] | |||
Revision as of 16:21, 19 August 2009
The abc conjecture asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed [math]\displaystyle{ c^{1-\varepsilon} }[/math] for any fixed [math]\displaystyle{ \varepsilon \gt 0 }[/math] (if a,b,c are smooth).
This shows for instance that [math]\displaystyle{ (1-\varepsilon) \log N / 3 }[/math]-smooth a,b,c of size N which are coprime cannot sum to form a+b=c. This unfortunately seems to be too weak to be of much use for the finding primes project.
